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Title: The Dirichlet problem with boundary data of fractional smoothness in domains with Ahlfors regular boundaries
Abstract: In recent years, numerous authors have investigated the Dirichlet problem for second order elliptic differential equations, with boundary data in Lebesgue or Sobolev spaces, in domains with very rough boundaries. In particular, some authors have considered the Dirichlet problem in domains with higher codimensional boundary. In this talk we will extrapolate their results to the Dirichlet problem with boundary data in Besov spaces. This is joint work with Svitlana Mayboroda and Alberto Pacati.
Title: Alberti representations, rectifiability and regularity of measures satisfying a PDE
Abstract: An Alberti representation of a (finite) measure is a decomposition into 1-rectifiable measures. By Fubini's theorem, Lebesgue measure on [0,1]^n has n "independent" Alberti representations, each one consisting of curves parallel to a coordinate axis. This notion naturally extends to Alberti representations of Hausdorff measure restricted to an n-rectifiable set in Euclidean space, where the tangents to curves span the approximate tangent space almost everywhere, and can be extended further to rectifiable subsets of a metric space. This talk will consider the converse statement. We will show that, if n-dimensional Hausdorff measure of a metric space X has n independent Alberti representations, then X is n-rectifiable. This is a strengthening of a previous result of Bate and Li where the conclusion was established under the additional hypothesis that X has positive lower n-density almost everywhere. This result has numerous applications to rectifiability in metric spaces that will be discussed. We will also discuss a key element of the proof that generalizes previous work of De Philippis and Rindler. Suppose S, T are two matrix valued measures on Euclidean space such that S has a constant and invertible polar and T has finite divergence. Then, provided S, T are sufficiently close in total variation, both have a large part that lies in L^p.
Title: Parabolic Uniform Rectifiability and Caloric measure
Abstract: Over the past four years, my coauthors and I have extended the David-Semmes theory of uniformly rectifiable sets to the parabolic setting. By defining these sets as those that are Ahlfors regular with a geometric square function in terms of "Jones beta numbers," we have established characterizations similar to those found in the pioneering work of David and Semmes. However, we have also identified several classical characterizations that do not hold in the parabolic context. Recently, our research has focused on exploring the connections between parabolic potential theory and parabolic uniform rectifiability. In this talk, I will present these results and, time permitting, discuss some open problems in this area.
Title: Venetian blinds, digital sundials, and efficient coverings
Abstract: Davies's efficient covering theorem states that we can cover any measurable set in the plane by lines without increasing the total measure. This result has a dual formulation, known as Falconer's digital sundial theorem, which states that we can construct a set in the plane to have any desired projections, up to null sets. The argument relies on a Venetian blind construction, a classical method in geometric measure theory. In joint work with Alex McDonald and Krystal Taylor, we study a variant of Davies's efficient covering theorem in which we replace lines with curves. This has a dual formulation in terms of nonlinear projections.
Title: The non-convex gradient constrained problem
Abstract: The problem
min(−∆u, |Du| − 1) = 0
arises as the Hamilton-Jacobi equation of a zero-sum game and can be thought as an elementary prototype of models in risk theory. It could be considered an obstacle problem with a constraint over the gradient, presenting new and intriguing questions in the analysis of nonlinear elliptic equations. In this presentation, I will introduce the model, provide the optimal regularity result for u and |Du| (in collaboration with E. Pimentel), offer a connection with a recently studied transmission problem, and discuss various aspects of the free boundary.
Title: Favard length problem for Ahlfors regular sets
Abstract: Favard length of a set is the average length of its orthogonal projections. The Besicovitch projection theorem states the following: for any set E of finite length whose Favard length is positive there exists a Lipschitz graph intersecting E in a set of positive length. The Favard length problem consists of quantifying this theorem, which is crucial for understanding the relation between Favard length and analytic capacity. In this talk I will discuss some recent work on this subject.
Title: Fractional parabolic theory as a high-dimensional limit of fractional elliptic theory
Abstract: Experts have long realized the parallels between elliptic and parabolic theory of partial differential equations. It is well-known that elliptic theory may be considered a static, or steady-state, version of parabolic theory. And in particular, if a parabolic estimate holds, then by eliminating the time parameter, one immediately arrives at the underlying elliptic statement. Producing a parabolic statement from an elliptic statement is not as straightforward. In this talk, we discuss how a high-dimensional limiting technique can be used to prove theorems about solutions to the fractional heat equation (or its Caffarelli-Silvestre extension problem) from their elliptic analogues. This talk covers joint work with Mariana Smit Vega Garcia.
Title: Absolute continuity of the Robin harmonic measure on rough domains
Abstract: This is joint work with S. Decio, M. Engelstein, S. Mayboroda, and M. Michetti. We prove that the analogue of elliptic measure, but with the Robin boundary condition, is mutually absolutely continuous with respect to surface or Hausdorff measure, for general elliptic operators L = - div(A $ and (to be checked) domains with one-sided NTA access and an Ahlfors regular boundary of dimension $d \in (n-2,n)$.
Title: Min-max construction of anisotropic minimal hypersurfaces
Abstract: We prove the existence of closed optimally regular hypersurfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed n-dimensional Riemannian manifolds. This proves a conjecture posed by Allard in 1983. The talk is based on joint work with Guido De Philippis and Yangyang Li.
Title: A unique continuation result for area minimizing currents
Abstract: Can two minimal surfaces touch each other to infinite order at a point without coinciding in a neighborhood of the point? Intuition from the theory of unique continuation for elliptic PDEs suggests this should not happen. Of course, part of the game here is to specify the notion of minimal surface. In joint work with Camillo Brena we give an answer to an instance of the question above: if an m-dimensional area minimizing integral current has infinite order of contact at a point with an m-dimensional surface with zero mean curvature then the current coincides with the surface in a neighborhood of the point.
Title: Regularity of capillary minimal hypersurfaces
Abstract: We describe two recent results concerning the regularity of minimal capillary hypersurfaces, i.e. surfaces meeting a container at prescribed angles. The first, joint with Otis Chodosh and Chao Li, is an improved dimension bound on the singular set for minimizing capillary hypersurfaces. We show the singular set has codimension at least 4, and this estimate improves for capillary angles close to $0$, $\pi/2$, or $\pi$. For small angles our analysis is based on a rigorous connection between the capillary problem and the one-phase Bernoulli problem. The second result, joint with Luigi de Masi, Carlo Gasparetto, and Chao Li, is an Allard-type regularity theorem for capillary surfaces which are merely stationary (or have bounded mean curvature). Using the notion of capillary varifold first introduced by Kagaya-Tonegawa, we prove a sharp first-variation bound for all angles, and prove that whenever the capillary varifold is close to a capillary half-plane of angle $\neq \pi/2$, then it coincides with a $C^{1,\alpha}$ hypersurface nearby. Our result implies regularity at generic boundary points of density $< 1$.
Title: Hausdorff dimension of caloric measure
Abstract: Caloric measure is a probability measure supported on the boundary of a domain in Rn+1=Rn×R (space × time) that is related to the Dirichlet problem for the heat equation in a fundamental way. Equipped with the parabolic distance, Rn+1 has Hausdorff dimension n+2. We prove that (even on domains with very large boundary), the caloric measure is carried by a set of Hausdorff dimension at most n+2-βn for some βn>0. The corresponding theorem for harmonic measure is due to Bourgain (1987), which states that the Hausdorff dimension of harmonic measure is at most n-bn for every domain in Rn. We refine Bourgain’s proof to obtain bn>cn-2n(n-1)/ln(n) for all n≥3, where c is independent of n. This is joint work with Matthew Badger.
Title: Self-similar sets and Lipschitz graphs
Abstract: We investigate and quantify the distinction between rectifiable and purely unrectifiable 1-sets in the plane. That is, given that purely unrectifiable 1-sets always have null intersections with Lipschitz images, we ask whether these sets intersect with Lipschitz images at a dimension that is close to one. In an answer to this question, we show that one-dimensional attractors of iterated function systems that satisfy the open set condition have subsets of dimension arbitrarily close to one that can be covered by Lipschitz graphs. Moreover, the Lipschitz constant of such graphs depends explicitly on the difference between the dimension of the original set and the subset that intersects with the graph. This is joint work with Blair Davey and Bobby Wilson.
Title: Rectifiability and tangent measures in a rough Riemannian setting
Abstract: In the 1920s Besicovitch asked the question: what can one say about the structure of a set E in the plane, with the property that as the limit as r goes to 0 of H1(B(x,r) ∩ E)/(2r) is 1 for almost every x in E? In 1987 Preiss gave a complete answer to Besicovitch's density question in the Euclidean setting in groundbreaking work introducing tangent measures. This talk introduces ongoing work with Emily Casey, Tatiana Toro, and Bobby Wilson that provides a new extension of Preiss' techniques to a rough Riemannian setting, as well as applications including new characterizations of rectifiable Radon measures and further developing the connection between the geometry of measures and elliptic PDEs.
Title: Quantitative rectifiability in metric spaces
Abstract: The theory of quantitative rectifiability for Ahlfors regular subsets of Euclidean space was developed extensively by David and Semmes in the early 1990s. They proved, among many other things, the equivalence of Uniform Rectifiability (UR) and the Bi-lateral Weak Geometric Lemma (BWGL). The first condition being a natural quantitative version of rectifiability, the second, a quantitative condition measuring local Hausdorff approximations by affine subspaces. In this talk I will discuss joint work with D. Bate and R. Schul which characterise UR metric spaces in terms of the BWGL and various other conditions related to the Euclidean theory of UR. We will introduce the problem, its history and formulate our main results. A continuation and expansion of the talk will be given by R. Schul.
Title: Low dimensional Cantor sets with absolutely continuous harmonic measure
Abstract: The relationship between harmonic measure and surface measure of a domain is largely connected with the geometry of the domain itself. In many fractals (for example, in domains with relatively “large” boundaries, and outside self-similar “enough” Cantor sets), these measures are mutually singular, and in fact, have different dimensions. After recalling some of these results I will present joint work with G. David and A. Julia where we demonstrate examples where the exact opposite occurs: we construct Cantor-type sets in the plane that are Ahlfors regular (of small dimension) for which their associated harmonic measure and surface measure are bounded equivalent.
Title: Unique continuation for nonlocal dispersive equations
Abstract: We will discuss recent works on unique continuation for nonlocal nonlinear dispersive equations, including the Benjamin-Ono and water wave equations as well as variable coefficient versions of them. This describes joint works with Pilod, Ponce and Vega.
Title: Uniformly rectifiable metric spaces satisfy the weak constant density condition
Abstract: The weak constant density condition (WCD) is a quantitative regularity property introduced by David and Semmes in their foundational work on uniformly rectifiable subsets of Euclidean spaces. Roughly speaking, a metric space satisfies the WCD if in “most” balls, the space supports a measure for which all sub-balls with comparable radius have very similar density. In this talk, we discuss some ideas behind a proof that uniformly rectifiable metric spaces satisfy the WCD. This theorem gives a metric space analog of a Euclidean result of David and Semmes and a quantitative analog of Kircheim’s theorem on densities in rectifiable metric spaces.
Title: Improved generic regularity of minimizing hypersurfaces
Abstract: I will discuss recent and ongoing work with O. Chodosh, F. Schulze, and Z. Wang showing that minimizing hypersurfaces have, after a suitable perturbation, a smaller than codimension-7 singular set.
Title: Perturbation of elliptic operators in rough domains
Abstract: The solvability of the Dirichlet problem with singular data for divergence form elliptic operators with real-valued bounded coefficients is related to the integrability properties of the associated elliptic measure. Given an operator L0 in this class, one can then study the perturbations of L0 whose associated elliptic measures behave as the one for the original operator L0. Based on the pioneering work by Fefferman, Kenig, and Pipher, and continued by Milakis, Pipher and Toro, in this talk we consider perturbations on which the discrepancies of the associated matrices of coefficients satisfy Carleson measure conditions (with large or small constants) and work in a settings that go beyond the class of non-tangentially accessible domains.
Title: Recent Developments in Varifold Regularity in the Critical Case
Abstract: This talk will introduce some recent developments in varifold regularity. Due to problems involving higher multiplicity, the regularity of the support of an m-dimensional varifold is usually studied under a collection of assumptions (hereafter referred to as the Allard Regularity Regime) which say that the varifold is "very close" to a multiplicity one m-disc in a ball. In 1972, Allard proved his famous regularity result for supercritical mean curvature (i.e., the generalized mean curvature is integrable with power p>m) under these assumptions. Since simple examples show that under the Allard Regularity Regime even the most basic regularity fails in the subcritical case (i.e., p<m), the majority of interest surrounds the critical case, when the generalized mean curvature has p=m power integrability. This problem has received a lot of attention in the special case that m=2, where it is closely related to the study of the Willmore energy. This talk will focus on some recent advancements in the critical case for arbitrary dimension and arbitrary codimension. These new results are part of a larger project to study varifold regularity in the critical case by building a PDE theory for generalized weakly differentiable functions on varifolds as introduced by U. Menne. Note that because varifolds may have nontrivial topology, the space of these functions a priori fails to even be a vector space! If time permits, there may be some discussion of preliminary results (joint with S. Kolasinski and U. Menne) in this larger project.
Title: Asymptotics of maximum distance minimizers
Abstract: The Maximum Distance Problem asks to find the shortest curve whose r-neighborhood contains a given set. Such curves are called r-maximum distance minimizers. We explore the limiting behavior of r-maximum distance minimizers as well as the asymptotics of their 1-dimensional Hausdorff measures as r tends to zero. Of note, we obtain results involving objects of fractal nature. This talk is based on is joint work with Enrique Alvarado, Louisa Catalano, and Tomás Merchán.
Title: The Saint-Venant inequality and quantitative resolvent estimates for the Dirichlet Laplacian
Abstract: Among all cylindrical beams of a given material, those with circular cross sections are the most resistant to twisting forces. The general dimensional analogue of this fact is the Saint-Venant inequality, which says that balls have the largest “torsional rigidity” among subsets of Euclidean space with a fixed volume. We discuss recent results showing that for a given set E, the gap in the Saint-Venant inequality quantitatively controls the L^2 difference between solutions of the Poisson equation on E and on the nearest ball, for any Holder continuous right-hand side. We additionally prove quantitative closeness of all eigenfunctions of the Dirichlet Laplacian. This talk is based on joint work with Mark Allen and Dennis Kriventsov.
Title: The Dirichlet problem as the boundary of the Poisson problem
Abstract: In this talk we will describe a result about how solutions to the Dirichlet problem with boundary data in Lp, p>1 in rough domains may be sharply approximated by a family of solutions to certain corresponding inhomogeneous Poisson problems with 0 boundary data. We will see a connection between this approximation result and a characterization of the dual space to the space of functions with Lp-bounded (modified) non-tangential maximal function. These results may be thought of as deeper instances of the robust relationship between (singular) boundary value problems and the Poisson problem recently investigated in the literature. This is joint work with Mihalis Mourgoglou.
Title: Uniformly rectifiable metric spaces
Abstract: In their 1991 and 1993 foundational monographs, David and Semmes characterized uniform rectifiability for subsets of Euclidean space in a multitude of geometric and analytic ways. The fundamental geometric conditions can be naturally stated in any metric space and it has long been a question of how these concepts are related in this general setting. In joint work with D. Bate and M. Hyde, we prove their equivalence. Namely, we show the equivalence of Big Pieces of Lipschitz Images, Bi-lateral Weak Geometric Lemma and Corona Decomposition in any Ahlfors regular metric space. Loosely speaking, this gives a quantitative equivalence between having Lipschitz charts and approximations by nice spaces. After giving some background, we will explain the main theorems and outline some key steps in the proof (which will include a discussion of Reifenberg parameterizations). We will also mention some open questions. This talk is an expansion and continuation of Matthew Hyde's Tuesday talk.
Title: Hölder curves with wild tangent spaces
Abstract: Rademacher's theorem is a classical result in geometric measure theory, which states that every Lipschitz function defined on the unit interval is differentiable at almost every point of the interval. In light of this, it seems natural to ask if there is some sense of tangents for which a similar result is true for Hölder curves as well. In this talk, we present a notion of tangent to a set E in Euclidean space due to Badger and Lewis and show that an analogue of Rademacher's theorem holds for this sense of tangent. Specifically, we show that for every Lipschitz curve in Euclidean space, at almost every point the tangent space to the curve contains exactly one line through the origin. Then we construct Hölder curves of arbitrary exponent below 1 in Euclidean space with positive and finite Hausdorff measure for which at almost every point, the tangent space contains infinitely many homeomorphically distinct elements. We conclude by noting that at almost every point, these curves possess at least one tangent that features a form of self-similarity, which leads us to a conjecture relating these tangent spaces to a Hölder version of the analyst's traveling salesman problem. This talk is based on joint work with Vyron Vellis.
Title: Flat singularities for area almost-minimizing currents
Abstract: The interior regularity of area-minimizing integral currents and semi-calibrated currents has been studied extensively in recent decades, with sharp dimension estimates and structural results established on their interior singular sets in any dimension and codimension. In stark contrast, the best partial regularity result for general almost-minimizing integral currents is that due to Bombieri in the 1980s, which demonstrates that the interior regular set is dense. We provide a construction of two types of examples that exhibit the sharpness of Bombieri’s result, and the dramatic failure of the regularity theory developed for area-minimizing integral currents and semi-calibrated integral currents. One type of example is a superposition of C1,α graphs meeting only tangentially at flat singularities, while the other is higher codimension, with genuinely branched flat singularities. Given any closed, empty interior subset K of an m-dimensional plane, we can produce an example of each type that has a flat singular set containing K. This is a joint work with Max Goering.
Title: Dimension of the singular set of 2-valued stationary graphs
Abstract: As a consequence of the celebrated Allard’s epsilon regularity theorem, it is well known that the singular set of an integral stationary n-varifold is meager. However, all known examples suggest that the Hausdorff dimension of such singular set should be n-1. In this talk I will present a recent result, joint with J. Hirsch (Leipzig), where we show that if the stationary varifold is a 2-valued Lipschitz n-graph, then indeed its singular set is of dimension n-1. I will spend ample time introducing the problem and explaining the main difficulties.
Title: Surfaces of minimum mean curvature variation
Abstract: We develop an analytic theory of existence and regularity of surfaces arising from a geometric minimization problem involving the quadratic variation of the mean curvature. The minimizers, called surfaces of minimum mean curvature variation, are central in applications of computer-aided design, computer-aided manufacturing and mechanics. We present the existence of both smooth surfaces and of variational solutions to the minimization problem together with sharp geometric regularity results. These are the first analytic results available on the literature for this problem. This is joint work with Luis A. Caffarelli (UT Austin) and Hern\’an Vivas (U. Nacional de Mar del Plata, Argentina).
Title: Solvability of the Neumann problem for elliptic PDE's in chord-arc domains with very big pieces of good superdomains
Abstract: One of the big open problems in the area of elliptic PDE's consists in showing that the Neumann problem for the Laplacian is solvable in L^p in chord-arc domains for some p>1. In this talk I will explain a recent result related to this question obtained in collaboration with Mihalis Mourgoglou where we show that, given an elliptic operator L and 1<p< 2, if Ω is a chord-arc domain in the Euclidean space such that the regularity problem for L is solvable in L^q for some q>p, and Ω has very big pieces of superdomains for which the Neumann problem is solvable uniformly in L^q, then the Neumann problem in Ω is solvable in L^p.
Title: Asymptotic expansion of a nonlocal phase transition energy
Abstract: In this talk, we first present the notion of asymptotic development in the sense of Γ-convergence for the fractional Allen-Cahn energy functional in bounded domains with prescribed boundary conditions. When the fractional power s in (0,1/2), we establish the complete asymptotic development by showing that the first-order term is the nonlocal minimal surface functional and all higher-order terms are zero. Then, for s in [1,2/1), we prove that the first-order term is the classical perimeter functional plus a penalization on the boundary. Consequently, we obtain an asymptotic expansion of the minimum values of the fractional Allen-Cahn energy.
Title: Carleson ε^2 conjecture in higher dimension and Faber-Krahn inequalities
Abstract: In this talk, I will report on a joint work with Ian Fleschler and Xavier Tolsa on higher dimensional analogues of the Carlesons ε^2 conjecture. In particular, we characterize tangent points of certain domains in Euclidean space via a novel "spherical" square function. Beyond its intrinsic geometric appeal, this result is motivated by connections to quantitative Faber-Krahn inequalities.
Title: Boundary unique continuation of harmonic functions
Abstract: Unique continuation property is a fundamental property for harmonic functions, as well as a large class of elliptic and parabolic PDEs. It says that if a harmonic function vanishes at a point to infinite order, it must vanish everywhere. In the same spirit, we are interested in quantitative unique continuation problems, where we use the local growth rate of a harmonic function to deduce some global estimates, such as estimating the size of its singular or critical set. In this talk, I will talk about some recent results together with C. Kenig on boundary unique continuation.
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